2020/05/16 by Peter Dankelmann, Dankelmann, Peter, Audace A. V. Dossou-Olory +1 · 1 citation
Mathematics · Chemistry · #Graph theory and applications #Synthesis and Properties of Aromatic Compounds #Zeolite Catalysis and Synthesis
paper · pdf · doi:10.48550/arxiv.2005.09462
The eccentric sequence of a connected graph G is the nondecreasing sequence\nof the eccentricities of its vertices. The Wiener index of G is the sum of\nthe distances between all unordered pairs of vertices of G. The unique trees\nthat minimise the Wiener index among all trees with a given eccentric sequence\nwere recently determined by the present authors. In this paper we show that\nthese results hold not only for the Wiener index, but for a large class of\ndistance-based topological indices which we term Wiener-type indices.\nParticular cases of this class include the hyper-Wiener index, the Harary\nindex, the generalised Wiener index W\λ for \λ>0 and \λ\n<0, and the reciprocal complementary Wiener index. Our results imply and unify\nknown bounds on these Wiener-type indices for trees of given order and\ndiameter.\n We also present similar results for the k-Steiner Wiener index of trees\nwith a given eccentric sequence. The Steiner distance of a set A\⊆\nV(G) is theminimum number of edges in a subtree of G whose vertex set\ncontains A, and the k-Steiner Wiener index is the sum of distances of all\nk-element subsets of V(G). As a corollary, we obtain a sharp lower bound on\nthe k-Steiner Wiener index of trees with given order and diameter, and\ndetermine in which cases the extremal tree is unique, thereby correcting an\nerror in the literature.\n